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Abstract

We introduce and study the notion of \emph{semiadditive height} for higher
semiadditive $\infty$-categories, which generalizes the chromatic height. We
show that the higher semiadditive structure trivializes above the height and
prove a form of the redshift principle, in which categorification increases the
height by one. In the stable setting, we show that a higher semiadditive
$\infty$-category decomposes into a product according to height, and relate the
notion of height to semisimplicity properties of local systems. We place the
study of higher semiadditivity and stability in the general framework of
smashing localizations of $Pr^{L}$, which we call \emph{modes}. Using this
theory, we introduce and study the universal stable $\infty$-semiadditive
$\infty$-category of semiadditive height $n$, and give sufficient conditions
for a stable $1$-semiadditive $\infty$-category to be $\infty$-semiadditive.